How to: Exponential Fall Off for Volume Density?
You can use Vector maths and Maths nodes to calculate the 'altitude' above the planet and from there the atmospheric density and use this to control the density of the Volumetric Scatter.

To achieve this, add a mesh around your planet to act as a domain for the scattering. The domain's centre should correspond with the centre of the spherical 'planet'.



Create the following material :



This uses vector maths to determine the distance from the centre of the planet. The Planet Radius value node determines where the atmosphere starts - it should correspond to the radius of the 'planet' object.

The 'Falloff Factor' determines how quickly the density of the atmosphere falls off - it should be negative and values closer to zero will provide a 'deeper' atmosphere (extending further from the surface of the planet).

The RGB node controls the 'color' of the atmosphere and the Multiply just prior to the scatter controls its 'thickness' at sea-level.

(Don't forget to increase the number of Volumetric bounces in the Cycles Light Paths settings.)

This can produce the following result :



Blend file attached 

Note : I've also created a material using and Emission shader instead of scatter - faking the planetary shadow - and this is far more efficient than using Scatter. The vector maths are far more complicated than in the above material so I've omitted it here for simplicity. I can provide additional details if it would be of interest.



Breaking down the nodes to explain this a little, we're starting with two coordinates - the centre of the planet (which is the same as the centre of the domain mesh) is provided by the Object Info node, and the point in space (where we want to determine the density) is provided by the Geometry node. Both of these are in 'World Space' coordinates (points relative to the centre of the world).



By subtracting one from the other we get the vector between the two points. The length of that vector represents the distance from the centre of the planet to the point in space and the Dot Product and Power nodes convert this into an absolute distance (rather than a vector). [Using the Dot product in this way (with the same vector connected to both inputs) produces the square of the distance of the vector. The Power(0.5) performs a square root, generating the actual distance.]

Next we subtract the radius of the planet. This will result in the actual altitude above the surface.



For the exponential falloff we need to use a Power node with the the first input set to the exponent (in this case '2') and the other input set to the altitude. For a 'falloff' we need a negative value and the Multiply node provides this as well as allowing the effect to be scaled. The final Multiply allows the overall density of the atmosphere to be adjusted.



As an example, if the Multiply node is set to $-1$, at the surface of the planet (0 altitude) the density would be :

$2 ^ 0 = 1$ (since anything to the power of 0 is 1)

At 1 blender unit above the surface,

$2 ^ {-1} = 0.5$

At 2 blender units above the surface,

$2 ^ {-2} = 0.25$

At 3 blender units above the surface,

$2 ^ {-3} = 0.125$

ie, The density falls off exponentially.

Adjusting the Falloff Factor to larger negative values will give a faster falloff.



EDIT: With your sample Blend file the patterns you are seeing are a result of rounding errors in the exponential falloff being amplified by the 250 multiplication factor (you have an atmosphere that is dropping off very sharply conflicting with it being very dense). Using your sample Blend file as a start point, perform the following steps :


Scale the Earth Atmo mesh by a factor of 2
Remove the unnecessary Subdivision Surface modifier
In the Earth Atmo shader, change the Multiply factor from 250 to 2
Change the division factor from 0.0012557 to 0.012557
In the Render settings change from Path Tracing to Branched Path Tracing (this renders the volumetrics better)


This should then produce the following result :